# Full text of "Treatise On Analysis Vol-Ii"

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1    EXISTENCE AND UNIQUENESS OF HAAR MEASURE       245

Properties (i), (ii), and (iii) are immediate consequences of the definitions.
Given (14.1.5.1), there exists (3.17.10) s e G such that

sup /(x) = /CO £ Z cig(sr 's) ^ ( Z c*) sup g(x),

xeG                       i=l                    \i=l    / jceG

and (iv) follows.  For  (v),  observe that if we have /^ZaiY(X')# anc*

9 = Z */Y(0)^> then/g £ aibjy(sitj)h, and therefore

J                                          i, J

i,j

Since we may take Z#i (resp. J^bj) arbitrarily close to (/: g) (resp. (g: h)),
we have (v). Finally, (vi) follows from (v) applied to/0, / g and to / f0, g.

By hypothesis, there exists a denumerable fundamental system (Vn) of
neighborhoods of the neutral element e in G. For each n, \ttgn be a function
belonging to «2f* such that Supp(#rt) <=zVn (4.5.2). Let/0 be a function in
Jf * , fixed once and for all, and put

(14.1.5.2)                         !„(/) = (/: gn)j(fQ : gn)

for all fe tf * ; let In(0) = 0. From (ii) and (iii) above it follows immedi-
ately that the mappings /-*!„( |/|) are seminorms on ^TR(G), and from (i)
that ln(y(s)f) = !„(/) for all s e G.

Next, there exists a sequence of relatively compact open sets Up which
cover G and are such that Op c Up+1 (3.18.3). The space ^(Up) is separable
(7.4.4) and therefore so is ^f(G; Op) n Jf * (3.10.9). Hence there exists a
dense sequence (fmp)m*i in the latter space. Since the sequence of values
ln(fmp) (n ^ 1) lies in the compact interval with endpoints l/(/0 :fmp) and
(fmp :/o) by (vi), it follows from (12.5.9) that if we replace the sequence (gn) by
a suitably chosen subsequence, we may assume that the sequence (ln(fmp))nzi
tends to a limit >Qfor all m, p. Also it is clear that if/,/' in Jf" * are such
that/^/', then !„(/) :g !„(/')• Let hp be a continuous mapping of G into
[0, 1] with compact support, taking the value 1 on Op ((3.18.2) and (4.5.2)).
Then, for each function/e jf (G; Op) n Jf * , we have !„(/) g ||/|| \n(hp) for
all /i, and therefore

for all//' e ^f(G, Up) n ^f + and all «, because/i-^In( |/j) is a seminorm.
In other words, the set of restrictions of the !„ to each of the subspaces
JT(G, Up) n jf ^ is equicontinuous. By (7.5.5) it follows that lim !„(/) = !(/)

»-+oo

exists for all/6 Jf *, and takes values ^0.
is called the Bernoulli scheme B(j?0 , • • • > Pm- 1).